###################### #Below is code to look for normal elements of degree 2 in the #families of algebras presented in Section 3.3. #It is not very efficient, so it takes a little while to run. # #The solution list on which the program is run contains in order #the solutions, A(b,q), B(b), C(b), D(1, h), E(1, \gamma), #\$F(1, \gamma), G(1, \gamma). Note that we only solve for b = 1 #in the examples D-G because this is sufficient in the #proof that those families are regular in the paper. # #The program finds all of the degree 2 normal elements claimed #in the paper. Note that the program finds some normal elements #which are normal only for special examples in the family, not #for general values of the parameters. So the program prints the #values of the parameters along with each solution. # #In examples D, E, G, once one mods out by degree 2 normal #elements, checking that the remainder of the elements in the #claimed normalizing sequence are normal is routine and can #easily be done by hand; we did not adjust the program to #do this. # #Note that the program finds no normal elements of degree 2 #in example F. This is the hardest example to deal with and we #have a separate program which finds normal elements of higher #degree for that example. ###################### rr1d3_sub_list:= {seq(x[r,2,1]= p*x[r,1,2], r = 1..3)} union {seq(x[2,1,s]= p*x[1,2,s], s = 1..3)}; rr2d3_sub_list:={seq(x[r,3,2] = a*x[r,3,1] + x[r,1,3] + b*x[r,2,3], r =1..3)} union {seq(x[3,2,s] = a*x[3,1,s] + x[1,3,s] + b*x[2,3,s], s = 1..3)}; rr3d3_sub_list:={x[3,3,1] = c*x[1,3,3] + d*x[2,3,3]+e*x[3,1,3]}; rr4d3_sub_list:= {x[3,1,1] = f*x[1,1,3] + g*x[1,2,3] +h*x[2,2,3] + j*x[1,3,1] + k*x[2,3,1]}; rr5d3_sub_list:= {x[3,1,2] = q*x[1,3,1] + q*b*x[2,3,1] + a*q*x[3,1,1]}; sol_sub_seq:=[{j = e, a = a, p = -1-e*a, c = (1+e*a)/a^2, h = 0, g = 0,d = 0, k = 0, b = -(2*e*a+e^2*a^2+1)/a, q = -1/(1+e*a), e = e, f =(1+e*a)/a^2},{q = -1, b = -1/a, c = 1/a^2, a = a, j = -1/a, h = 0, p = -1, f = 0, d = 0, e = 0, k = 1/a^2, g = 1/a^3}, {q = -1, b=-1/a, c = 1/a^2, a = a, j = -1/a, h = 0, p = -1, f = 0, d = 0, e =0, k = 1/a^2, g = -1/a^3},{c = 1, q = -1, p = -1, d = 0, e = 0, g = 0, k = 0, j = 0, h = h, a = -1, f = -1+h, b = 1}, {q = -1, h = 0, p = -1, f = 0, e = 0, c = 0, j = 1, d = 1, g = RootOf(_Z^2+1), k = 1, a = -1, b = 1}, {e = -1, h = 0, f = 0, j = 0, c = 0, q = RootOf(_Z^2+_Z+1), p = -1-RootOf(_Z^2+_Z+1), a = -RootOf(_Z^2+_Z+1), d = -1-RootOf(_Z^2+_Z+1), b = 1, g = RootOf(_Z^2+_Z+1), k = -1-RootOf(_Z^2+_Z+1)}, {q = -1, p = -1, e = 0, g = 0, k = 0, j = 0, c = 0, d = 1, a = -1, b = 1, h = RootOf(2*_Z^2-2*_Z+1), f = 1-RootOf(2*_Z^2-2*_Z+1)}]; for m from 1 by 1 to nops(sol_sub_seq) do gone_sub_list2[m]:={Y[m,3,2]=0, Y[m,2,1]=0}; r1d3_sub_list[m]:=subs(sol_sub_seq[m], rr1d3_sub_list); r2d3_sub_list[m]:=subs(sol_sub_seq[m], rr2d3_sub_list); r3d3_sub_list[m]:=subs(sol_sub_seq[m], rr3d3_sub_list); r4d3_sub_list[m]:=subs(sol_sub_seq[m], rr4d3_sub_list); r5d3_sub_list[m]:=subs(sol_sub_seq[m], rr5d3_sub_list); for r from 1 by 1 to 3 do PP[m,r]:=add(add(Y[m,t,u]*x[r,t,u], u=1..3), t=1..3) end do; for r from 1 by 1 to 3 do QQ[m,r]:=add(add(add(Y[m,t,u]*W[m,v,r]*x[t,u,v], u=1..3), t=1..3), v=1..3) end do; for r from 1 by 1 to 3 do PP1[m,r]:=subs(gone_sub_list2[m], PP[m,r]) end do; for r from 1 by 1 to 3 do QQ1[m,r]:=subs(gone_sub_list2[m], QQ[m,r]) end do; for r from 1 by 1 to 3 do RR1[m,r]:=PP1[m,r]-QQ1[m,r] end do; for r from 1 by 1 to 3 do RR2[m,r]:=subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], RR1[m,r]))))) end do; for r from 1 by 1 to 3 do RR3[m,r]:=subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], RR2[m,r]))))) end do; for r from 1 by 1 to 3 do RR4[m,r]:=subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], RR3[m,r]))))) end do; for r from 1 by 1 to 3 do RR5[m,r]:=subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m],subs(r1d3_sub_list[m], RR4[m,r]))))) end do; for r from 1 by 1 to 3 do RR[m,r]:=subs(r5d3_sub_list[m], subs(r4d3_sub_list[m], subs(r3d3_sub_list[m], subs(r2d3_sub_list[m], subs(r1d3_sub_list[m], RR5[m,r]))))) end do; Ylist[m]:={seq(seq(Y[m,t,u], t=1..3), u=1..3)}; Wlist[m]:={seq(seq(W[m,r,v], r=1..3), v=1..3)}; for v from 1 by 1 to 3 do VV[m,v]:= {seq(seq(seq(simplify(coeff(RR[m,v], x[r,t,u])), r = 1..3), t = 1..3), u = 1..3)} end do; SS[m]:=solve(VV[m,1] union VV[m,2] union VV[m,3] union sol_sub_seq[m], {a,b,c,d,e,q,p,f,g,h,j,k} union Ylist[m] union Wlist[m]); TT1[m]:=subs(gone_sub_list2[m], add(add(Y[m,t,u]*x[t,u], t=1..3), u=1..3)); for v from 1 by 1 to 3 do TT2[m,v]:=add(W[m,r,v]*x[r], r =1..3) end do end do; for m from 1 by 1 to nops(sol_sub_seq) do print("_______________________________________________"); print("_______________________________________________"); print("finding normal elements for solution number", m); print("_______________________________________________"); print(sol_sub_seq[m]); print("_______________________________________________"); for u from 1 by 1 to nops({SS[m]}) do print("deg 2 normal element", u); NE2[m,u]:=simplify(subs({SS[m]}[u], TT1[m])); if NE2[m,u]<>0 then print("normal element=",NE2[m,u]); for v from 1 by 1 to 3 do print(x[v], "*", NE2[m,u], "=", NE2[m,u], "*", simplify(subs({SS[m]}[u], TT2[m,v]))) end do; print("a=", simplify(subs({SS[m]}[u], a)), "b=", simplify(subs({SS[m]}[u], b)), "c=", simplify(subs({SS[m]}[u], c)), "d=", simplify(subs({SS[m]}[u], d)), "e=", simplify(subs({SS[m]}[u], e)), "f=", simplify(subs({SS[m]}[u], f)), "g=", simplify(subs({SS[m]}[u], g)), "h=", simplify(subs({SS[m]}[u], h)), "j=", simplify(subs({SS[m]}[u], j)), "k=", simplify(subs({SS[m]}[u], k)), "q=", simplify(subs({SS[m]}[u], q))); print("-------------------------------------------------------------------") end if end do end do; ####################### #Final output from above program: ###################### "_______________________________________________" "_______________________________________________" "finding normal elements for solution number", 1 "_______________________________________________" 2 2 2 e a + e a + 1 1 + e a {a = a, b = - -----------------, c = -------, d = 0, e = e, a 2 a 1 + e a f = -------, g = 0, h = 0, j = e, k = 0, p = -1 - e a, 2 a 1 q = - -------} 1 + e a "_______________________________________________" "deg 2 normal element", 1 "normal element=", Y[1, 2, 2] x[1, 1] + Y[1, 3, 1] x[3, 1] + 2 Y[1, 2, 2] x[1, 2] + Y[1, 2, 2] x[2, 2] - Y[1, 3, 1] x[1, 3] x[1], "*", Y[1, 2, 2] x[1, 1] + Y[1, 3, 1] x[3, 1] + 2 Y[1, 2, 2] x[1, 2] + Y[1, 2, 2] x[2, 2] - Y[1, 3, 1] x[1, 3], "=", Y[1, 2, 2] x[1, 1] + Y[1, 3, 1] x[3, 1] + 2 Y[1, 2, 2] x[1, 2] + Y[1, 2, 2] x[2, 2] - Y[1, 3, 1] x[1, 3], "*", x[1] x[2], "*", Y[1, 2, 2] x[1, 1] + Y[1, 3, 1] x[3, 1] + 2 Y[1, 2, 2] x[1, 2] + Y[1, 2, 2] x[2, 2] - Y[1, 3, 1] x[1, 3], "=", Y[1, 2, 2] x[1, 1] + Y[1, 3, 1] x[3, 1] + 2 Y[1, 2, 2] x[1, 2] + Y[1, 2, 2] x[2, 2] - Y[1, 3, 1] x[1, 3], "*", x[2] x[3], "*", Y[1, 2, 2] x[1, 1] + Y[1, 3, 1] x[3, 1] + 2 Y[1, 2, 2] x[1, 2] + Y[1, 2, 2] x[2, 2] - Y[1, 3, 1] x[1, 3], "=", Y[1, 2, 2] x[1, 1] + Y[1, 3, 1] x[3, 1] + 2 Y[1, 2, 2] x[1, 2] + Y[1, 2, 2] x[2, 2] - Y[1, 3, 1] x[1, 3], "*", x[3] "a=", -1, "b=", 1, "c=", -1, "d=", 0, "e=", 2, "f=", -1, "g=", 0, "h=", 0, "j=", 2, "k=", 0, "q=", 1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 2 "normal element=", Y[1, 1, 1] x[1, 1] + Y[1, 2, 2] x[2, 2] + Y[1, 3, 3] x[3, 3] x[1], "*", Y[1, 1, 1] x[1, 1] + Y[1, 2, 2] x[2, 2] + Y[1, 3, 3] x[3, 3], "=", Y[1, 1, 1] x[1, 1] + Y[1, 2, 2] x[2, 2] + Y[1, 3, 3] x[3, 3], "*", x[1] x[2], "*", Y[1, 1, 1] x[1, 1] + Y[1, 2, 2] x[2, 2] + Y[1, 3, 3] x[3, 3], "=", Y[1, 1, 1] x[1, 1] + Y[1, 2, 2] x[2, 2] + Y[1, 3, 3] x[3, 3], "*", x[2] x[3], "*", Y[1, 1, 1] x[1, 1] + Y[1, 2, 2] x[2, 2] + Y[1, 3, 3] x[3, 3], "=", Y[1, 1, 1] x[1, 1] + Y[1, 2, 2] x[2, 2] + Y[1, 3, 3] x[3, 3], "*", x[3] "a=", -1, "b=", 1, "c=", 1, "d=", 0, "e=", 0, "f=", 1, "g=", 0, "h=", 0, "j=", 0, "k=", 0, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 3 "normal element=", Y[1, 1, 1] x[1, 1] + Y[1, 3, 1] x[3, 1] + Y[1, 2, 2] x[2, 2] + Y[1, 3, 1] x[1, 3] + Y[1, 3, 3] x[3, 3] x[1], "*", Y[1, 1, 1] x[1, 1] + Y[1, 3, 1] x[3, 1] + Y[1, 2, 2] x[2, 2] + Y[1, 3, 1] x[1, 3] + Y[1, 3, 3] x[3, 3], "=", Y[1, 1, 1] x[1, 1] + Y[1, 3, 1] x[3, 1] + Y[1, 2, 2] x[2, 2] + Y[1, 3, 1] x[1, 3] + Y[1, 3, 3] x[3, 3], "*", x[1] x[2], "*", Y[1, 1, 1] x[1, 1] + Y[1, 3, 1] x[3, 1] + Y[1, 2, 2] x[2, 2] + Y[1, 3, 1] x[1, 3] + Y[1, 3, 3] x[3, 3], "=", Y[1, 1, 1] x[1, 1] + Y[1, 3, 1] x[3, 1] + Y[1, 2, 2] x[2, 2] + Y[1, 3, 1] x[1, 3] + Y[1, 3, 3] x[3, 3], "*", x[2] x[3], "*", Y[1, 1, 1] x[1, 1] + Y[1, 3, 1] x[3, 1] + Y[1, 2, 2] x[2, 2] + Y[1, 3, 1] x[1, 3] + Y[1, 3, 3] x[3, 3], "=", Y[1, 1, 1] x[1, 1] + Y[1, 3, 1] x[3, 1] + Y[1, 2, 2] x[2, 2] + Y[1, 3, 1] x[1, 3] + Y[1, 3, 3] x[3, 3], "*", x[3] "a=", 1, "b=", -1, "c=", 1, "d=", 0, "e=", 0, "f=", 1, "g=", 0, "h=", 0, "j=", 0, "k=", 0, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 4 "normal element=", Y[1, 1, 1] x[1, 1] + Y[1, 2, 2] x[2, 2] x[1], "*", Y[1, 1, 1] x[1, 1] + Y[1, 2, 2] x[2, 2], "=", Y[1, 1, 1] x[1, 1] + Y[1, 2, 2] x[2, 2], "*", x[1] x[2], "*", Y[1, 1, 1] x[1, 1] + Y[1, 2, 2] x[2, 2], "=", Y[1, 1, 1] x[1, 1] + Y[1, 2, 2] x[2, 2], "*", x[2] x[3], "*", Y[1, 1, 1] x[1, 1] + Y[1, 2, 2] x[2, 2], "=", x[3] Y[1, 1, 1] x[1, 1] + Y[1, 2, 2] x[2, 2], "*", ---- 2 a 1 1 "a=", a, "b=", - 1/a, "c=", ----, "d=", 0, "e=", 0, "f=", ----, 2 2 a a "g=", 0, "h=", 0, "j=", 0, "k=", 0, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 5 "normal element=", Y[1, 3, 3] x[3, 3] x[1], "*", Y[1, 3, 3] x[3, 3], "=", Y[1, 3, 3] x[3, 3], "*", 2 a x[1] x[2], "*", Y[1, 3, 3] x[3, 3], "=", Y[1, 3, 3] x[3, 3], "*", 2 a x[2] x[3], "*", Y[1, 3, 3] x[3, 3], "=", Y[1, 3, 3] x[3, 3], "*", x[3] 1 1 "a=", a, "b=", - 1/a, "c=", ----, "d=", 0, "e=", 0, "f=", ----, 2 2 a a "g=", 0, "h=", 0, "j=", 0, "k=", 0, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 6 Y[1, 3, 1] (x[3, 1] a + x[1, 3]) "normal element=", -------------------------------- a Y[1, 3, 1] (x[3, 1] a + x[1, 3]) x[1], "*", --------------------------------, "=", a Y[1, 3, 1] (x[3, 1] a + x[1, 3]) x[1] --------------------------------, "*", ---------- a W[1, 3, 3] Y[1, 3, 1] (x[3, 1] a + x[1, 3]) x[2], "*", --------------------------------, "=", a Y[1, 3, 1] (x[3, 1] a + x[1, 3]) x[2] --------------------------------, "*", ---------- a W[1, 3, 3] Y[1, 3, 1] (x[3, 1] a + x[1, 3]) x[3], "*", --------------------------------, "=", a Y[1, 3, 1] (x[3, 1] a + x[1, 3]) --------------------------------, "*", W[1, 3, 3] x[3] a 2 W[1, 3, 3] "a=", a, "b=", -a W[1, 3, 3] , "c=", ----------, "d=", 0, "e=", a -1 + W[1, 3, 3] a W[1, 3, 3] -----------------, "f=", ----------, "g=", 0, "h=", 0, "j=", a a -1 + W[1, 3, 3] a 1 -----------------, "k=", 0, "q=", - ------------ a W[1, 3, 3] a "---------------------------------------------------------------\ ----" "deg 2 normal element", 7 "deg 2 normal element", 8 "normal element=", 2 Y[1, 2, 2] (4 x[1, 1] + 4 x[1, 2] e + x[2, 2] e ) ------------------------------------------------- 2 e 2 Y[1, 2, 2] (4 x[1, 1] + 4 x[1, 2] e + x[2, 2] e ) x[1], "*", -------------------------------------------------, "=", 2 e 2 Y[1, 2, 2] (4 x[1, 1] + 4 x[1, 2] e + x[2, 2] e ) -------------------------------------------------, "*", x[1] 2 e 2 Y[1, 2, 2] (4 x[1, 1] + 4 x[1, 2] e + x[2, 2] e ) x[2], "*", -------------------------------------------------, "=", 2 e 2 Y[1, 2, 2] (4 x[1, 1] + 4 x[1, 2] e + x[2, 2] e ) -------------------------------------------------, "*", x[2] 2 e 2 Y[1, 2, 2] (4 x[1, 1] + 4 x[1, 2] e + x[2, 2] e ) x[3], "*", -------------------------------------------------, "=", 2 e 2 Y[1, 2, 2] (4 x[1, 1] + 4 x[1, 2] e + x[2, 2] e ) -------------------------------------------------, "*", 2 e 2 1/4 e x[3] 2 e "a=", - 2/e, "b=", e/2, "c=", - ----, "d=", 0, "e=", e, "f=", 4 2 e - ----, "g=", 0, "h=", 0, "j=", e, "k=", 0, "q=", 1 4 "---------------------------------------------------------------\ ----" "deg 2 normal element", 9 "normal element=", Y[1, 3, 1] x[3, 1] - 1/2 Y[1, 3, 1] e x[1, 3] x[1], "*", Y[1, 3, 1] x[3, 1] - 1/2 Y[1, 3, 1] e x[1, 3], "=", 2 x[1] Y[1, 3, 1] x[3, 1] - 1/2 Y[1, 3, 1] e x[1, 3], "*", ------ e x[2], "*", Y[1, 3, 1] x[3, 1] - 1/2 Y[1, 3, 1] e x[1, 3], "=", 2 x[2] Y[1, 3, 1] x[3, 1] - 1/2 Y[1, 3, 1] e x[1, 3], "*", ------ e x[3], "*", Y[1, 3, 1] x[3, 1] - 1/2 Y[1, 3, 1] e x[1, 3], "=", Y[1, 3, 1] x[3, 1] - 1/2 Y[1, 3, 1] e x[1, 3], "*", 1/2 e x[3] 2 e "a=", - 2/e, "b=", e/2, "c=", - ----, "d=", 0, "e=", e, "f=", 4 2 e - ----, "g=", 0, "h=", 0, "j=", e, "k=", 0, "q=", 1 4 "---------------------------------------------------------------\ ----" "deg 2 normal element", 10 "_______________________________________________" "_______________________________________________" "finding normal elements for solution number", 2 "_______________________________________________" 1 1 {a = a, b = - 1/a, c = ----, d = 0, e = 0, f = 0, g = ----, h = 0, 2 3 a a 1 j = - 1/a, k = ----, p = -1, q = -1} 2 a "_______________________________________________" "deg 2 normal element", 1 "normal element=", Y[2, 3, 1] x[3, 1] - Y[2, 3, 1] x[2, 3] x[1], "*", Y[2, 3, 1] x[3, 1] - Y[2, 3, 1] x[2, 3], "=", Y[2, 3, 1] x[3, 1] - Y[2, 3, 1] x[2, 3], "*", x[1] x[2], "*", Y[2, 3, 1] x[3, 1] - Y[2, 3, 1] x[2, 3], "=", Y[2, 3, 1] x[3, 1] - Y[2, 3, 1] x[2, 3], "*", x[2] x[3], "*", Y[2, 3, 1] x[3, 1] - Y[2, 3, 1] x[2, 3], "=", Y[2, 3, 1] x[3, 1] - Y[2, 3, 1] x[2, 3], "*", x[3] "a=", -1, "b=", 1, "c=", 1, "d=", 0, "e=", 0, "f=", 0, "g=", -1, "h=", 0, "j=", 1, "k=", 1, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 2 "normal element=", Y[2, 1, 1] x[1, 1] - Y[2, 2, 3] x[3, 1] + Y[2, 1, 1] x[2, 2] + Y[2, 2, 3] x[2, 3] + Y[2, 3, 3] x[3, 3] x[1], "*", Y[2, 1, 1] x[1, 1] - Y[2, 2, 3] x[3, 1] + Y[2, 1, 1] x[2, 2] + Y[2, 2, 3] x[2, 3] + Y[2, 3, 3] x[3, 3], "=", Y[2, 1, 1] x[1, 1] - Y[2, 2, 3] x[3, 1] + Y[2, 1, 1] x[2, 2] + Y[2, 2, 3] x[2, 3] + Y[2, 3, 3] x[3, 3], "*", x[1] x[2], "*", Y[2, 1, 1] x[1, 1] - Y[2, 2, 3] x[3, 1] + Y[2, 1, 1] x[2, 2] + Y[2, 2, 3] x[2, 3] + Y[2, 3, 3] x[3, 3], "=", Y[2, 1, 1] x[1, 1] - Y[2, 2, 3] x[3, 1] + Y[2, 1, 1] x[2, 2] + Y[2, 2, 3] x[2, 3] + Y[2, 3, 3] x[3, 3], "*", x[2] x[3], "*", Y[2, 1, 1] x[1, 1] - Y[2, 2, 3] x[3, 1] + Y[2, 1, 1] x[2, 2] + Y[2, 2, 3] x[2, 3] + Y[2, 3, 3] x[3, 3], "=", Y[2, 1, 1] x[1, 1] - Y[2, 2, 3] x[3, 1] + Y[2, 1, 1] x[2, 2] + Y[2, 2, 3] x[2, 3] + Y[2, 3, 3] x[3, 3], "*", x[3] "a=", -1, "b=", 1, "c=", 1, "d=", 0, "e=", 0, "f=", 0, "g=", -1, "h=", 0, "j=", 1, "k=", 1, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 3 "normal element=", -Y[2, 2, 3] x[3, 1] + Y[2, 1, 2] x[1, 2] + Y[2, 2, 3] x[2, 3] x[1], "*", -Y[2, 2, 3] x[3, 1] + Y[2, 1, 2] x[1, 2] + Y[2, 2, 3] x[2, 3] , "=", -Y[2, 2, 3] x[3, 1] + Y[2, 1, 2] x[1, 2] + Y[2, 2, 3] x[2, 3] , "*", -x[1] x[2], "*", -Y[2, 2, 3] x[3, 1] + Y[2, 1, 2] x[1, 2] + Y[2, 2, 3] x[2, 3] , "=", -Y[2, 2, 3] x[3, 1] + Y[2, 1, 2] x[1, 2] + Y[2, 2, 3] x[2, 3] , "*", -x[2] x[3], "*", -Y[2, 2, 3] x[3, 1] + Y[2, 1, 2] x[1, 2] + Y[2, 2, 3] x[2, 3] , "=", -Y[2, 2, 3] x[3, 1] + Y[2, 1, 2] x[1, 2] + Y[2, 2, 3] x[2, 3] , "*", -x[3] "a=", 1, "b=", -1, "c=", 1, "d=", 0, "e=", 0, "f=", 0, "g=", 1, "h=", 0, "j=", -1, "k=", 1, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 4 "normal element=", Y[2, 1, 1] x[1, 1] + Y[2, 1, 1] x[2, 2] + Y[2, 3, 3] x[3, 3] x[1], "*", Y[2, 1, 1] x[1, 1] + Y[2, 1, 1] x[2, 2] + Y[2, 3, 3] x[3, 3], "=", Y[2, 1, 1] x[1, 1] + Y[2, 1, 1] x[2, 2] + Y[2, 3, 3] x[3, 3], "*", x[1] x[2], "*", Y[2, 1, 1] x[1, 1] + Y[2, 1, 1] x[2, 2] + Y[2, 3, 3] x[3, 3], "=", Y[2, 1, 1] x[1, 1] + Y[2, 1, 1] x[2, 2] + Y[2, 3, 3] x[3, 3], "*", x[2] x[3], "*", Y[2, 1, 1] x[1, 1] + Y[2, 1, 1] x[2, 2] + Y[2, 3, 3] x[3, 3], "=", Y[2, 1, 1] x[1, 1] + Y[2, 1, 1] x[2, 2] + Y[2, 3, 3] x[3, 3], "*", x[3] "a=", 1, "b=", -1, "c=", 1, "d=", 0, "e=", 0, "f=", 0, "g=", 1, "h=", 0, "j=", -1, "k=", 1, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 5 "normal element=", Y[2, 1, 2] x[1, 2] x[1], "*", Y[2, 1, 2] x[1, 2], "=", Y[2, 1, 2] x[1, 2], "*", -x[1] x[2], "*", Y[2, 1, 2] x[1, 2], "=", Y[2, 1, 2] x[1, 2], "*", -x[2] x[3] x[3], "*", Y[2, 1, 2] x[1, 2], "=", Y[2, 1, 2] x[1, 2], "*", - ---- 2 a 1 "a=", a, "b=", - 1/a, "c=", ----, "d=", 0, "e=", 0, "f=", 0, "g=", 2 a 1 1 ----, "h=", 0, "j=", - 1/a, "k=", ----, "q=", -1 3 2 a a "---------------------------------------------------------------\ ----" "deg 2 normal element", 6 2 Y[2, 1, 1] (x[1, 1] a + x[2, 2]) "normal element=", --------------------------------- 2 a 2 Y[2, 1, 1] (x[1, 1] a + x[2, 2]) x[1], "*", ---------------------------------, "=", 2 a 2 Y[2, 1, 1] (x[1, 1] a + x[2, 2]) ---------------------------------, "*", x[1] 2 a 2 Y[2, 1, 1] (x[1, 1] a + x[2, 2]) x[2], "*", ---------------------------------, "=", 2 a 2 Y[2, 1, 1] (x[1, 1] a + x[2, 2]) ---------------------------------, "*", x[2] 2 a 2 Y[2, 1, 1] (x[1, 1] a + x[2, 2]) x[3], "*", ---------------------------------, "=", 2 a 2 Y[2, 1, 1] (x[1, 1] a + x[2, 2]) x[3] ---------------------------------, "*", ---- 2 2 a a 1 "a=", a, "b=", - 1/a, "c=", ----, "d=", 0, "e=", 0, "f=", 0, "g=", 2 a 1 1 ----, "h=", 0, "j=", - 1/a, "k=", ----, "q=", -1 3 2 a a "---------------------------------------------------------------\ ----" "deg 2 normal element", 7 "deg 2 normal element", 8 "normal element=", Y[2, 3, 3] x[3, 3] x[1], "*", Y[2, 3, 3] x[3, 3], "=", Y[2, 3, 3] x[3, 3], "*", 2 a x[1] x[2], "*", Y[2, 3, 3] x[3, 3], "=", Y[2, 3, 3] x[3, 3], "*", 2 a x[2] x[3], "*", Y[2, 3, 3] x[3, 3], "=", Y[2, 3, 3] x[3, 3], "*", x[3] 1 "a=", a, "b=", - 1/a, "c=", ----, "d=", 0, "e=", 0, "f=", 0, "g=", 2 a 1 1 ----, "h=", 0, "j=", - 1/a, "k=", ----, "q=", -1 3 2 a a "---------------------------------------------------------------\ ----" "deg 2 normal element", 9 2 "normal element=", -Y[2, 2, 3] a x[3, 1] + Y[2, 2, 3] x[2, 3] 2 x[1], "*", -Y[2, 2, 3] a x[3, 1] + Y[2, 2, 3] x[2, 3], "=", 2 -Y[2, 2, 3] a x[3, 1] + Y[2, 2, 3] x[2, 3], "*", -a x[1] 2 x[2], "*", -Y[2, 2, 3] a x[3, 1] + Y[2, 2, 3] x[2, 3], "=", 2 -Y[2, 2, 3] a x[3, 1] + Y[2, 2, 3] x[2, 3], "*", -a x[2] 2 x[3], "*", -Y[2, 2, 3] a x[3, 1] + Y[2, 2, 3] x[2, 3], "=", 2 x[3] -Y[2, 2, 3] a x[3, 1] + Y[2, 2, 3] x[2, 3], "*", - ---- a 1 "a=", a, "b=", - 1/a, "c=", ----, "d=", 0, "e=", 0, "f=", 0, "g=", 2 a 1 1 ----, "h=", 0, "j=", - 1/a, "k=", ----, "q=", -1 3 2 a a "---------------------------------------------------------------\ ----" "deg 2 normal element", 10 2 Y[2, 2, 3] (-x[3, 1] + x[2, 3] W[2, 3, 3] ) "normal element=", ------------------------------------------- 2 W[2, 3, 3] 2 Y[2, 2, 3] (-x[3, 1] + x[2, 3] W[2, 3, 3] ) x[1], "*", -------------------------------------------, "=", 2 W[2, 3, 3] 2 Y[2, 2, 3] (-x[3, 1] + x[2, 3] W[2, 3, 3] ) x[1] -------------------------------------------, "*", ---------- 2 W[2, 3, 3] W[2, 3, 3] 2 Y[2, 2, 3] (-x[3, 1] + x[2, 3] W[2, 3, 3] ) x[2], "*", -------------------------------------------, "=", 2 W[2, 3, 3] 2 Y[2, 2, 3] (-x[3, 1] + x[2, 3] W[2, 3, 3] ) x[2] -------------------------------------------, "*", ---------- 2 W[2, 3, 3] W[2, 3, 3] 2 Y[2, 2, 3] (-x[3, 1] + x[2, 3] W[2, 3, 3] ) x[3], "*", -------------------------------------------, "=", 2 W[2, 3, 3] 2 Y[2, 2, 3] (-x[3, 1] + x[2, 3] W[2, 3, 3] ) -------------------------------------------, "*", 2 W[2, 3, 3] W[2, 3, 3] x[3] 1 2 "a=", - ----------, "b=", W[2, 3, 3], "c=", W[2, 3, 3] , "d=", 0, W[2, 3, 3] 3 "e=", 0, "f=", 0, "g=", -W[2, 3, 3] , "h=", 0, "j=", 2 W[2, 3, 3], "k=", W[2, 3, 3] , "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 11 "deg 2 normal element", 12 "normal element=", Y[2, 3, 3] x[3, 3] x[1], "*", Y[2, 3, 3] x[3, 3], "=", Y[2, 3, 3] x[3, 3], "*", -x[1] x[2], "*", Y[2, 3, 3] x[3, 3], "=", Y[2, 3, 3] x[3, 3], "*", -x[2] x[3], "*", Y[2, 3, 3] x[3, 3], "=", Y[2, 3, 3] x[3, 3], "*", x[3] "a=", %1, "b=", %1, "c=", -1, "d=", 0, "e=", 0, "f=", 0, "g=", %1, "h=", 0, "j=", %1, "k=", -1, "q=", -1 2 %1 := RootOf(_Z + 1, label = _L221) "---------------------------------------------------------------\ ----" "deg 2 normal element", 13 "normal element=", Y[2, 1, 2] x[1, 2] x[1], "*", Y[2, 1, 2] x[1, 2], "=", Y[2, 1, 2] x[1, 2], "*", -x[1] x[2], "*", Y[2, 1, 2] x[1, 2], "=", Y[2, 1, 2] x[1, 2], "*", -x[2] x[3], "*", Y[2, 1, 2] x[1, 2], "=", Y[2, 1, 2] x[1, 2], "*", x[3] "a=", %1, "b=", %1, "c=", -1, "d=", 0, "e=", 0, "f=", 0, "g=", %1, "h=", 0, "j=", %1, "k=", -1, "q=", -1 2 %1 := RootOf(_Z + 1, label = _L222) "---------------------------------------------------------------\ ----" "deg 2 normal element", 14 "deg 2 normal element", 15 "normal element=", Y[2, 1, 2] x[1, 2] x[1], "*", Y[2, 1, 2] x[1, 2], "=", Y[2, 1, 2] x[1, 2], "*", -x[1] x[2], "*", Y[2, 1, 2] x[1, 2], "=", Y[2, 1, 2] x[1, 2], "*", -x[2] x[3], "*", Y[2, 1, 2] x[1, 2], "=", Y[2, 1, 2] x[1, 2], "*", x[3] "a=", %1, "b=", %1, "c=", -1, "d=", 0, "e=", 0, "f=", 0, "g=", %1, "h=", 0, "j=", %1, "k=", -1, "q=", -1 2 %1 := RootOf(_Z + 1, label = _L224) "---------------------------------------------------------------\ ----" "deg 2 normal element", 16 "normal element=", Y[2, 3, 3] x[3, 3] x[1], "*", Y[2, 3, 3] x[3, 3], "=", Y[2, 3, 3] x[3, 3], "*", -x[1] x[2], "*", Y[2, 3, 3] x[3, 3], "=", Y[2, 3, 3] x[3, 3], "*", -x[2] x[3], "*", Y[2, 3, 3] x[3, 3], "=", Y[2, 3, 3] x[3, 3], "*", x[3] "a=", %1, "b=", %1, "c=", -1, "d=", 0, "e=", 0, "f=", 0, "g=", %1, "h=", 0, "j=", %1, "k=", -1, "q=", -1 2 %1 := RootOf(_Z + 1, label = _L225) "---------------------------------------------------------------\ ----" "deg 2 normal element", 17 "normal element=", Y[2, 1, 2] x[1, 2] + Y[2, 3, 3] x[3, 3] x[1], "*", Y[2, 1, 2] x[1, 2] + Y[2, 3, 3] x[3, 3], "=", Y[2, 1, 2] x[1, 2] + Y[2, 3, 3] x[3, 3], "*", -x[1] x[2], "*", Y[2, 1, 2] x[1, 2] + Y[2, 3, 3] x[3, 3], "=", Y[2, 1, 2] x[1, 2] + Y[2, 3, 3] x[3, 3], "*", -x[2] x[3], "*", Y[2, 1, 2] x[1, 2] + Y[2, 3, 3] x[3, 3], "=", Y[2, 1, 2] x[1, 2] + Y[2, 3, 3] x[3, 3], "*", x[3] "a=", %1, "b=", %1, "c=", -1, "d=", 0, "e=", 0, "f=", 0, "g=", %1, "h=", 0, "j=", %1, "k=", -1, "q=", -1 2 %1 := RootOf(_Z + 1, label = _L226) "---------------------------------------------------------------\ ----" "deg 2 normal element", 18 "_______________________________________________" "_______________________________________________" "finding normal elements for solution number", 3 "_______________________________________________" 1 1 {a = a, b = - 1/a, c = ----, d = 0, e = 0, f = 0, g = - ----, h = 0, 2 3 a a 1 j = - 1/a, k = ----, p = -1, q = -1} 2 a "_______________________________________________" "deg 2 normal element", 1 "normal element=", Y[3, 1, 1] x[1, 1] + Y[3, 1, 1] x[2, 2] + Y[3, 3, 3] x[3, 3] x[1], "*", Y[3, 1, 1] x[1, 1] + Y[3, 1, 1] x[2, 2] + Y[3, 3, 3] x[3, 3], "=", Y[3, 1, 1] x[1, 1] + Y[3, 1, 1] x[2, 2] + Y[3, 3, 3] x[3, 3], "*", x[1] x[2], "*", Y[3, 1, 1] x[1, 1] + Y[3, 1, 1] x[2, 2] + Y[3, 3, 3] x[3, 3], "=", Y[3, 1, 1] x[1, 1] + Y[3, 1, 1] x[2, 2] + Y[3, 3, 3] x[3, 3], "*", x[2] x[3], "*", Y[3, 1, 1] x[1, 1] + Y[3, 1, 1] x[2, 2] + Y[3, 3, 3] x[3, 3], "=", Y[3, 1, 1] x[1, 1] + Y[3, 1, 1] x[2, 2] + Y[3, 3, 3] x[3, 3], "*", x[3] "a=", -1, "b=", 1, "c=", 1, "d=", 0, "e=", 0, "f=", 0, "g=", 1, "h=", 0, "j=", 1, "k=", 1, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 2 "normal element=", Y[3, 1, 1] x[1, 1] + Y[3, 1, 1] x[2, 2] + Y[3, 3, 3] x[3, 3] x[1], "*", Y[3, 1, 1] x[1, 1] + Y[3, 1, 1] x[2, 2] + Y[3, 3, 3] x[3, 3], "=", Y[3, 1, 1] x[1, 1] + Y[3, 1, 1] x[2, 2] + Y[3, 3, 3] x[3, 3], "*", x[1] x[2], "*", Y[3, 1, 1] x[1, 1] + Y[3, 1, 1] x[2, 2] + Y[3, 3, 3] x[3, 3], "=", Y[3, 1, 1] x[1, 1] + Y[3, 1, 1] x[2, 2] + Y[3, 3, 3] x[3, 3], "*", x[2] x[3], "*", Y[3, 1, 1] x[1, 1] + Y[3, 1, 1] x[2, 2] + Y[3, 3, 3] x[3, 3], "=", Y[3, 1, 1] x[1, 1] + Y[3, 1, 1] x[2, 2] + Y[3, 3, 3] x[3, 3], "*", x[3] "a=", 1, "b=", -1, "c=", 1, "d=", 0, "e=", 0, "f=", 0, "g=", -1, "h=", 0, "j=", -1, "k=", 1, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 3 "normal element=", Y[3, 1, 2] x[1, 2] x[1], "*", Y[3, 1, 2] x[1, 2], "=", Y[3, 1, 2] x[1, 2], "*", -x[1] x[2], "*", Y[3, 1, 2] x[1, 2], "=", Y[3, 1, 2] x[1, 2], "*", -x[2] x[3] x[3], "*", Y[3, 1, 2] x[1, 2], "=", Y[3, 1, 2] x[1, 2], "*", ---- 2 a 1 "a=", a, "b=", - 1/a, "c=", ----, "d=", 0, "e=", 0, "f=", 0, "g=", 2 a 1 1 - ----, "h=", 0, "j=", - 1/a, "k=", ----, "q=", -1 3 2 a a "---------------------------------------------------------------\ ----" "deg 2 normal element", 4 Y[3, 3, 1] (x[3, 1] a + x[1, 3]) "normal element=", -------------------------------- a Y[3, 3, 1] (x[3, 1] a + x[1, 3]) x[1], "*", --------------------------------, "=", a Y[3, 3, 1] (x[3, 1] a + x[1, 3]) --------------------------------, "*", x[2] a Y[3, 3, 1] (x[3, 1] a + x[1, 3]) x[2], "*", --------------------------------, "=", a Y[3, 3, 1] (x[3, 1] a + x[1, 3]) 2 --------------------------------, "*", a x[1] a Y[3, 3, 1] (x[3, 1] a + x[1, 3]) x[3], "*", --------------------------------, "=", a Y[3, 3, 1] (x[3, 1] a + x[1, 3]) x[3] --------------------------------, "*", ---- a a 1 "a=", a, "b=", - 1/a, "c=", ----, "d=", 0, "e=", 0, "f=", 0, "g=", 2 a 1 1 - ----, "h=", 0, "j=", - 1/a, "k=", ----, "q=", -1 3 2 a a "---------------------------------------------------------------\ ----" "deg 2 normal element", 5 2 Y[3, 1, 1] (x[1, 1] a + x[2, 2]) "normal element=", --------------------------------- 2 a 2 Y[3, 1, 1] (x[1, 1] a + x[2, 2]) x[1], "*", ---------------------------------, "=", 2 a 2 Y[3, 1, 1] (x[1, 1] a + x[2, 2]) ---------------------------------, "*", x[1] 2 a 2 Y[3, 1, 1] (x[1, 1] a + x[2, 2]) x[2], "*", ---------------------------------, "=", 2 a 2 Y[3, 1, 1] (x[1, 1] a + x[2, 2]) ---------------------------------, "*", x[2] 2 a 2 Y[3, 1, 1] (x[1, 1] a + x[2, 2]) x[3], "*", ---------------------------------, "=", 2 a 2 Y[3, 1, 1] (x[1, 1] a + x[2, 2]) x[3] ---------------------------------, "*", ---- 2 2 a a 1 "a=", a, "b=", - 1/a, "c=", ----, "d=", 0, "e=", 0, "f=", 0, "g=", 2 a 1 1 - ----, "h=", 0, "j=", - 1/a, "k=", ----, "q=", -1 3 2 a a "---------------------------------------------------------------\ ----" "deg 2 normal element", 6 "deg 2 normal element", 7 "normal element=", Y[3, 3, 3] x[3, 3] x[1], "*", Y[3, 3, 3] x[3, 3], "=", Y[3, 3, 3] x[3, 3], "*", 2 a x[1] x[2], "*", Y[3, 3, 3] x[3, 3], "=", Y[3, 3, 3] x[3, 3], "*", 2 a x[2] x[3], "*", Y[3, 3, 3] x[3, 3], "=", Y[3, 3, 3] x[3, 3], "*", x[3] 1 "a=", a, "b=", - 1/a, "c=", ----, "d=", 0, "e=", 0, "f=", 0, "g=", 2 a 1 1 - ----, "h=", 0, "j=", - 1/a, "k=", ----, "q=", -1 3 2 a a "---------------------------------------------------------------\ ----" "deg 2 normal element", 8 "deg 2 normal element", 9 "normal element=", Y[3, 3, 3] x[3, 3] x[1], "*", Y[3, 3, 3] x[3, 3], "=", Y[3, 3, 3] x[3, 3], "*", -x[1] x[2], "*", Y[3, 3, 3] x[3, 3], "=", Y[3, 3, 3] x[3, 3], "*", -x[2] x[3], "*", Y[3, 3, 3] x[3, 3], "=", Y[3, 3, 3] x[3, 3], "*", x[3] "a=", %1, "b=", %1, "c=", -1, "d=", 0, "e=", 0, "f=", 0, "g=", -%1, "h=", 0, "j=", %1, "k=", -1, "q=", -1 2 %1 := RootOf(_Z + 1, label = _L250) "---------------------------------------------------------------\ ----" "deg 2 normal element", 10 "deg 2 normal element", 11 "normal element=", Y[3, 3, 3] x[3, 3] x[1], "*", Y[3, 3, 3] x[3, 3], "=", Y[3, 3, 3] x[3, 3], "*", -x[1] x[2], "*", Y[3, 3, 3] x[3, 3], "=", Y[3, 3, 3] x[3, 3], "*", -x[2] x[3], "*", Y[3, 3, 3] x[3, 3], "=", Y[3, 3, 3] x[3, 3], "*", x[3] "a=", %1, "b=", %1, "c=", -1, "d=", 0, "e=", 0, "f=", 0, "g=", -%1, "h=", 0, "j=", %1, "k=", -1, "q=", -1 2 %1 := RootOf(_Z + 1, label = _L255) "---------------------------------------------------------------\ ----" "deg 2 normal element", 12 "_______________________________________________" "_______________________________________________" "finding normal elements for solution number", 4 "_______________________________________________" {a = -1, b = 1, c = 1, d = 0, e = 0, f = -1 + h, g = 0, h = h, j = 0, k = 0, p = -1, q = -1} "_______________________________________________" "deg 2 normal element", 1 "normal element=", Y[4, 1, 1] x[1, 1] x[1], "*", Y[4, 1, 1] x[1, 1], "=", Y[4, 1, 1] x[1, 1], "*", x[1] x[2], "*", Y[4, 1, 1] x[1, 1], "=", Y[4, 1, 1] x[1, 1], "*", x[2] x[3], "*", Y[4, 1, 1] x[1, 1], "=", Y[4, 1, 1] x[1, 1], "*", -x[3] "a=", -1, "b=", 1, "c=", 1, "d=", 0, "e=", 0, "f=", -1, "g=", 0, "h=", 0, "j=", 0, "k=", 0, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 2 "normal element=", Y[4, 3, 1] x[3, 1] - Y[4, 3, 1] x[2, 3] x[1], "*", Y[4, 3, 1] x[3, 1] - Y[4, 3, 1] x[2, 3], "=", Y[4, 3, 1] x[3, 1] - Y[4, 3, 1] x[2, 3], "*", -x[2] x[2], "*", Y[4, 3, 1] x[3, 1] - Y[4, 3, 1] x[2, 3], "=", Y[4, 3, 1] x[3, 1] - Y[4, 3, 1] x[2, 3], "*", -x[1] x[3], "*", Y[4, 3, 1] x[3, 1] - Y[4, 3, 1] x[2, 3], "=", Y[4, 3, 1] x[3, 1] - Y[4, 3, 1] x[2, 3], "*", x[3] "a=", -1, "b=", 1, "c=", 1, "d=", 0, "e=", 0, "f=", 0, "g=", 0, "h=", 1, "j=", 0, "k=", 0, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 3 Y[4, 2, 2] (-2 x[1, 1] + x[1, 1] h + x[2, 2] h) "normal element=", ----------------------------------------------- h Y[4, 2, 2] (-2 x[1, 1] + x[1, 1] h + x[2, 2] h) x[1], "*", -----------------------------------------------, "=", h Y[4, 2, 2] (-2 x[1, 1] + x[1, 1] h + x[2, 2] h) -----------------------------------------------, "*", x[1] h Y[4, 2, 2] (-2 x[1, 1] + x[1, 1] h + x[2, 2] h) x[2], "*", -----------------------------------------------, "=", h Y[4, 2, 2] (-2 x[1, 1] + x[1, 1] h + x[2, 2] h) -----------------------------------------------, "*", x[2] h Y[4, 2, 2] (-2 x[1, 1] + x[1, 1] h + x[2, 2] h) x[3], "*", -----------------------------------------------, "=", h Y[4, 2, 2] (-2 x[1, 1] + x[1, 1] h + x[2, 2] h) -----------------------------------------------, "*", -x[3] h "a=", -1, "b=", 1, "c=", 1, "d=", 0, "e=", 0, "f=", -1 + h, "g=", 0, "h=", h, "j=", 0, "k=", 0, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 4 "normal element=", Y[4, 1, 1] x[1, 1] + Y[4, 1, 1] x[2, 2] + Y[4, 3, 3] x[3, 3] x[1], "*", Y[4, 1, 1] x[1, 1] + Y[4, 1, 1] x[2, 2] + Y[4, 3, 3] x[3, 3], "=", Y[4, 1, 1] x[1, 1] + Y[4, 1, 1] x[2, 2] + Y[4, 3, 3] x[3, 3], "*", x[1] x[2], "*", Y[4, 1, 1] x[1, 1] + Y[4, 1, 1] x[2, 2] + Y[4, 3, 3] x[3, 3], "=", Y[4, 1, 1] x[1, 1] + Y[4, 1, 1] x[2, 2] + Y[4, 3, 3] x[3, 3], "*", x[2] x[3], "*", Y[4, 1, 1] x[1, 1] + Y[4, 1, 1] x[2, 2] + Y[4, 3, 3] x[3, 3], "=", Y[4, 1, 1] x[1, 1] + Y[4, 1, 1] x[2, 2] + Y[4, 3, 3] x[3, 3], "*", x[3] "a=", -1, "b=", 1, "c=", 1, "d=", 0, "e=", 0, "f=", -1 + h, "g=", 0, "h=", h, "j=", 0, "k=", 0, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 5 "_______________________________________________" "_______________________________________________" "finding normal elements for solution number", 5 "_______________________________________________" 2 {a = -1, b = 1, c = 0, d = 1, e = 0, f = 0, g = RootOf(_Z + 1), h = 0, j = 1, k = 1, p = -1, q = -1} "_______________________________________________" "deg 2 normal element", 1 "normal element=", Y[5, 1, 2] x[1, 2] x[1], "*", Y[5, 1, 2] x[1, 2], "=", Y[5, 1, 2] x[1, 2], "*", -x[1] x[2], "*", Y[5, 1, 2] x[1, 2], "=", Y[5, 1, 2] x[1, 2], "*", -x[2] x[3], "*", Y[5, 1, 2] x[1, 2], "=", Y[5, 1, 2] x[1, 2], "*", 2 RootOf(_Z + 1) x[3] "a=", -1, "b=", 1, "c=", 0, "d=", 1, "e=", 0, "f=", 0, "g=", 2 RootOf(_Z + 1), "h=", 0, "j=", 1, "k=", 1, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 2 "normal element=", Y[5, 3, 3] x[3, 3] x[1], "*", Y[5, 3, 3] x[3, 3], "=", Y[5, 3, 3] x[3, 3], "*", x[2] x[2], "*", Y[5, 3, 3] x[3, 3], "=", Y[5, 3, 3] x[3, 3], "*", x[1] x[3], "*", Y[5, 3, 3] x[3, 3], "=", Y[5, 3, 3] x[3, 3], "*", x[3] "a=", -1, "b=", 1, "c=", 0, "d=", 1, "e=", 0, "f=", 0, "g=", 2 RootOf(_Z + 1), "h=", 0, "j=", 1, "k=", 1, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 3 "normal element=", Y[5, 2, 2] x[1, 1] + Y[5, 2, 2] x[2, 2] x[1], "*", Y[5, 2, 2] x[1, 1] + Y[5, 2, 2] x[2, 2], "=", Y[5, 2, 2] x[1, 1] + Y[5, 2, 2] x[2, 2], "*", x[1] x[2], "*", Y[5, 2, 2] x[1, 1] + Y[5, 2, 2] x[2, 2], "=", Y[5, 2, 2] x[1, 1] + Y[5, 2, 2] x[2, 2], "*", x[2] x[3], "*", Y[5, 2, 2] x[1, 1] + Y[5, 2, 2] x[2, 2], "=", Y[5, 2, 2] x[1, 1] + Y[5, 2, 2] x[2, 2], "*", x[3] "a=", -1, "b=", 1, "c=", 0, "d=", 1, "e=", 0, "f=", 0, "g=", 2 RootOf(_Z + 1), "h=", 0, "j=", 1, "k=", 1, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 4 "_______________________________________________" "_______________________________________________" "finding normal elements for solution number", 6 "_______________________________________________" {a = -%1, b = 1, c = 0, d = -1 - %1, e = -1, f = 0, g = %1, h = 0, j = 0, k = -1 - %1, p = -1 - %1, q = %1} 2 %1 := RootOf(_Z + _Z + 1) "_______________________________________________" "deg 2 normal element", 1 "_______________________________________________" "_______________________________________________" "finding normal elements for solution number", 7 "_______________________________________________" {a = -1, b = 1, c = 0, d = 1, e = 0, 2 f = 1 - RootOf(2 _Z - 2 _Z + 1), g = 0, 2 h = RootOf(2 _Z - 2 _Z + 1), j = 0, k = 0, p = -1, q = -1} "_______________________________________________" "deg 2 normal element", 1 "normal element=", Y[7, 3, 3] x[3, 3] x[1], "*", Y[7, 3, 3] x[3, 3], "=", Y[7, 3, 3] x[3, 3], "*", x[2] x[2], "*", Y[7, 3, 3] x[3, 3], "=", Y[7, 3, 3] x[3, 3], "*", x[1] x[3], "*", Y[7, 3, 3] x[3, 3], "=", Y[7, 3, 3] x[3, 3], "*", x[3] "a=", -1, "b=", 1, "c=", 0, "d=", 1, "e=", 0, "f=", 2 1 - RootOf(2 _Z - 2 _Z + 1), "g=", 0, "h=", 2 RootOf(2 _Z - 2 _Z + 1), "j=", 0, "k=", 0, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 2 "normal element=", Y[7, 1, 1] x[1, 1] + Y[7, 1, 1] x[2, 2] x[1], "*", Y[7, 1, 1] x[1, 1] + Y[7, 1, 1] x[2, 2], "=", Y[7, 1, 1] x[1, 1] + Y[7, 1, 1] x[2, 2], "*", x[1] x[2], "*", Y[7, 1, 1] x[1, 1] + Y[7, 1, 1] x[2, 2], "=", Y[7, 1, 1] x[1, 1] + Y[7, 1, 1] x[2, 2], "*", x[2] x[3], "*", Y[7, 1, 1] x[1, 1] + Y[7, 1, 1] x[2, 2], "=", Y[7, 1, 1] x[1, 1] + Y[7, 1, 1] x[2, 2], "*", x[3] "a=", -1, "b=", 1, "c=", 0, "d=", 1, "e=", 0, "f=", 2 1 - RootOf(2 _Z - 2 _Z + 1), "g=", 0, "h=", 2 RootOf(2 _Z - 2 _Z + 1), "j=", 0, "k=", 0, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 3 "normal element=", Y[7, 1, 1] x[1, 1] - Y[7, 1, 1] x[2, 2] x[1], "*", Y[7, 1, 1] x[1, 1] - Y[7, 1, 1] x[2, 2], "=", Y[7, 1, 1] x[1, 1] - Y[7, 1, 1] x[2, 2], "*", x[1] x[2], "*", Y[7, 1, 1] x[1, 1] - Y[7, 1, 1] x[2, 2], "=", Y[7, 1, 1] x[1, 1] - Y[7, 1, 1] x[2, 2], "*", x[2] x[3], "*", Y[7, 1, 1] x[1, 1] - Y[7, 1, 1] x[2, 2], "=", Y[7, 1, 1] x[1, 1] - Y[7, 1, 1] x[2, 2], "*", 2 -(2 RootOf(2 _Z - 2 _Z + 1) - 1) x[3] "a=", -1, "b=", 1, "c=", 0, "d=", 1, "e=", 0, "f=", 2 1 - RootOf(2 _Z - 2 _Z + 1), "g=", 0, "h=", 2 RootOf(2 _Z - 2 _Z + 1), "j=", 0, "k=", 0, "q=", -1 "---------------------------------------------------------------\ ----" "deg 2 normal element", 4 >